Set up an exponential function to model each real world situation then use it to find the requested values.

The half-life of Carbon 14 is about 5700 years. You begin with 800 grams of this isotope.

1) Function:

2) How much Carbon 14 is there after: a) 11,400 years b) 45,000 years

f(t) = 800(1/2)^(t/5700)

You can see that every time t increases by 5700, you have another power of 1/2, or a half-life.

So, using that, just plug in your values for t.

You can get an estimate by noting that

11400 = 2*5700
45000 ≈ 8*5700

To set up an exponential function to model the decay of Carbon 14, we can use the general form of an exponential function:

N(t) = N₀ * e^(kt)

Where:
- N(t) is the amount of Carbon 14 remaining after time t
- N₀ is the initial amount of Carbon 14
- k is the decay constant
- e is the base of the natural logarithm (~2.71828)

1) Function:
Since the half-life of Carbon 14 is about 5700 years, we can use this information to find the decay constant, k.

Half-life equation:
N(t) = N₀ * (1/2)^(t/h)

Where:
- h is the half-life

By substituting t = h and N(t) = N₀/2 into the equation, we can solve for k:

N₀/2 = N₀ * (1/2)^(h/k)

Simplifying the equation, we get:

(1/2)^(h/k) = 1/2
(1/2)^(h/k) = (1/2)^1

Since the bases are equal, the exponents must also be equal:

h/k = 1

Therefore, we have:

k = h

Now we can substitute this value of k in the exponential function:

N(t) = N₀ * e^(kt)
N(t) = N₀ * e^(ht)

For the given situation, let's set N₀ = 800 grams.

2) a) To find the amount of Carbon 14 after 11,400 years, we can use the exponential function:

N(t) = N₀ * e^(ht)

Substituting N₀ = 800 grams and t = 11,400 years:

N(11,400) = 800 * e^(5700 * 11,400)

You can use a calculator to calculate the exponential expression.

2) b) To find the amount of Carbon 14 after 45,000 years, we can again use the exponential function:

N(t) = N₀ * e^(ht)

Substituting N₀ = 800 grams and t = 45,000 years:

N(45,000) = 800 * e^(5700 * 45,000)

You can use a calculator to calculate the exponential expression.